13 ideas
| 18889 | Ostensive definitions needn't involve pointing, but must refer to something specific [Salmon,N] |
| Full Idea: So-called ostensive definitions need not literally involve ostension, e.g. pointing, but they must involve genuine reference of some sort (in this case reference to a sample of water). | |
| From: Nathan Salmon (Reference and Essence (1st edn) [1981], 4.11.2) |
| 14627 | S4, and therefore S5, are invalid for metaphysical modality [Salmon,N, by Williamson] |
| Full Idea: Salmon argues that S4 and therefore S5 are invalid for metaphysical modality. | |
| From: report of Nathan Salmon (Reference and Essence (1st edn) [1981], 238-40) by Timothy Williamson - Modal Logic within Counterfactual Logic 4 | |
| A reaction: [He gives references for Salmon, and for his own reply] Salmon's view seems to be opposed my most modern logicians (such as Ian Rumfitt). |
| 10185 | Set theory is the standard background for modern mathematics [Burgess] |
| Full Idea: In present-day mathematics, it is set theory that serves as the background theory in which other branches of mathematics are developed. | |
| From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §1) | |
| A reaction: [He cites Bourbaki as an authority for this] See Benacerraf for a famous difficulty here, when you actually try to derive an ontology from the mathematicians' working practices. |
| 10184 | Structuralists take the name 'R' of the reals to be a variable ranging over structures, not a structure [Burgess] |
| Full Idea: On the structuralist interpretation, theorems of analysis concerning the real numbers R are about all complete ordered fields. So R, which appears to be the name of a specific structure, is taken to be a variable ranging over structures. | |
| From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §1) | |
| A reaction: Since I am beginning to think that nearly all linguistic expressions should be understood as variables, I find this very appealing, even if Burgess hates it. Terms slide and drift, and are vague, between variable and determinate reference. |
| 10189 | There is no one relation for the real number 2, as relations differ in different models [Burgess] |
| Full Idea: One might meet the 'Van Inwagen Problem' by saying that the intrinsic properties of the object playing the role of 2 will differ from one model to another, so that no statement about the intrinsic properties of 'the' real numbers will make sense. | |
| From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §5) | |
| A reaction: There seems to be a potential confusion among opponents of structuralism between relations at the level of actual mathematical operations, and generalisations about relations, which are captured in the word 'patterns'. Call them 'meta-relations'? |
| 10186 | If set theory is used to define 'structure', we can't define set theory structurally [Burgess] |
| Full Idea: It is to set theory that one turns for the very definition of 'structure', ...and this creates a problem of circularity if we try to impose a structuralist interpretation on set theory. | |
| From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §1) | |
| A reaction: This seems like a nice difficulty, especially if, like Shapiro, you wade in and try to give a formal account of structures and patterns. Resnik is more circumspect and vague. |
| 10187 | Abstract algebra concerns relations between models, not common features of all the models [Burgess] |
| Full Idea: Abstract algebra, such as group theory, is not concerned with the features common to all models of the axioms, but rather with the relationships among different models of those axioms (especially homomorphic relation functions). | |
| From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §1) | |
| A reaction: It doesn't seem to follow that structuralism can't be about the relations (or patterns) found when abstracting away and overviewing all the models. One can study family relations, or one can study kinship in general. |
| 10188 | How can mathematical relations be either internal, or external, or intrinsic? [Burgess] |
| Full Idea: The 'Van Inwagen Problem' for structuralism is of explaining how a mathematical relation (such as set membership, or the ratios of an ellipse) can fit into one of the three scholastics types of relations: are they internal, external, or intrinsic? | |
| From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §5) | |
| A reaction: The difficulty is that mathematical objects seem to need intrinsic properties to get any of these three versions off the ground (which was Russell's complaint against structures). |
| 18888 | Essentialism says some properties must be possessed, if a thing is to exist [Salmon,N] |
| Full Idea: The metaphysical doctrine of essentialism says that certain properties of things are properties that those things could not fail to have, except by not existing. | |
| From: Nathan Salmon (Reference and Essence (1st edn) [1981], 3.8.2) | |
| A reaction: A bad account of essentialism, and a long way from Aristotle. It arises from the logicians' tendency to fix objects entirely in terms of a 'flat' list of predicates (called 'properties'!), which ignore structure, constitution, history etc. |
| 18886 | Frege's 'sense' solves four tricky puzzles [Salmon,N] |
| Full Idea: Reference via sense solves Frege's four puzzles, of the informativeness of identity statements, the failure of substitutivity in attitude contexts, of negative existentials, and the truth-value of statements using nondenoting singular terms. | |
| From: Nathan Salmon (Reference and Essence (1st edn) [1981], 1.1.1) | |
| A reaction: These must then be compared with Kripke's three puzzles about referring via sense, and the whole debate is then spread before us. |
| 18887 | The perfect case of direct reference is a variable which has been assigned a value [Salmon,N] |
| Full Idea: The paradigm of a nondescriptional, directly referential, singular term is an individual variable. …The denotation of a variable… is semantically determined directly by the assignment of values. | |
| From: Nathan Salmon (Reference and Essence (1st edn) [1981], 1.1.2) | |
| A reaction: This cuts both ways. Maybe we are muddling ordinary reference with the simplicities of logical assignments, or maybe we make logical assignments because that is the natural way our linguistic thinking works. |
| 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? |
| 18891 | Nothing in the direct theory of reference blocks anti-essentialism; water structure might have been different [Salmon,N] |
| Full Idea: There seems to be nothing in the theory of direct reference to block the anti-essentialist assertion that the substance water might have been the very same entity and yet have had a different chemical structure. | |
| From: Nathan Salmon (Reference and Essence (1st edn) [1981], 6.23.1) | |
| A reaction: Indeed, water could be continuously changing its inner structure, while retaining the surface appearance that gets baptised as 'water'. We make the reasonable empirical assumption, though, that structure-change implies surface-change. |