8 ideas
| 10882 | Predicative definitions only refer to entities outside the defined collection [Horsten] |
| Full Idea: Definitions are called 'predicative', and are considered sound, if they only refer to entities which exist independently from the defined collection. | |
| From: Leon Horsten (Philosophy of Mathematics [2007], §2.4) |
| 10884 | A theory is 'categorical' if it has just one model up to isomorphism [Horsten] |
| Full Idea: If a theory has, up to isomorphism, exactly one model, then it is said to be 'categorical'. | |
| From: Leon Horsten (Philosophy of Mathematics [2007], §5.2) |
| 10885 | Computer proofs don't provide explanations [Horsten] |
| Full Idea: Mathematicians are uncomfortable with computerised proofs because a 'good' proof should do more than convince us that a certain statement is true. It should also explain why the statement in question holds. | |
| From: Leon Horsten (Philosophy of Mathematics [2007], §5.3) |
| 10881 | The concept of 'ordinal number' is set-theoretic, not arithmetical [Horsten] |
| Full Idea: The notion of an ordinal number is a set-theoretic, and hence non-arithmetical, concept. | |
| From: Leon Horsten (Philosophy of Mathematics [2007], §2.3) |
| 9073 | Abstraction from an ambiguous concept like 'mole' will define them as the same [Barnes,J] |
| Full Idea: The procedure of abstraction will not allow us to distinguish the ambiguity between 'mole' as an animal and as an artefact. The stages of abstraction will only end up with 'physical object', and this will then count as the definition. | |
| From: Jonathan Barnes (Commentary on 'Posterior Analytics [1993], n to 97b7) | |
| A reaction: This is a problem if you adhere to a rather precise account of the steps of abstraction, with every stage explicit (and probably expressed in terms of sets), but I suspect that the real tangle of semi-conscious abstraction avoids this problem. |
| 9074 | Abstraction cannot produce the concept of a 'game', as there is no one common feature [Barnes,J] |
| Full Idea: Abstractions cannot account for those general terms whose instances do not have any set of features in common. The word 'game' is not ambiguous, but not all games have one thing in common; they are united by looser 'family resemblance'. | |
| From: Jonathan Barnes (Commentary on 'Posterior Analytics [1993], n to 97b7) | |
| A reaction: (This point comes from Wittgenstein, Idea 4141) English-speakers can't agree on borderline cases (avoiding cracks in pavements). Life is just a game. The objection would be refuted by discussion of higher-level abstractions to make connections. |
| 9072 | Defining concepts by abstractions will collect together far too many attributes from entities [Barnes,J] |
| Full Idea: If we create abstractions by collection of attributes common to groups of entities, we will collect far too many attributes, and wrongly put them into the definition (such as 'having hairless palms' when identifying 'men'). | |
| From: Jonathan Barnes (Commentary on 'Posterior Analytics [1993], n to 97b7) | |
| A reaction: [compressed] Defining 'man' is a hugely complex business (see Idea 1763!), unlike defining 'hair' or 'red'. Some attributes will strike perceivers immediately, but absence of an attribute is not actually 'perceived' at all. |
| 19216 | Propositions (such as 'that dog is barking') only exist if their items exist [Williamson] |
| Full Idea: A proposition about an item exists only if that item exists... how could something be the proposition that that dog is barking in circumstances in which that dog does not exist? | |
| From: Timothy Williamson (Necessary Existents [2002], p.240), quoted by Trenton Merricks - Propositions | |
| A reaction: This is a view of propositions I can't make sense of. If I'm under an illusion that there is a dog barking nearby, when there isn't one, can I not say 'that dog is barking'? If I haven't expressed a proposition, what have I done? |