11 ideas
| 15457 | Interdefinition is useless by itself, but if we grasp one separately, we have them both [Lewis] |
| Full Idea: All circles of interdefinition are useless by themselves. But if we reach one of the interdefined pair, then we have them both. | |
| From: David Lewis (Defining 'Intrinsic' (with Rae Langton) [1998], IV) |
| 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). |
| 15400 | We must avoid circularity between what is intrinsic and what is natural [Lewis, by Cameron] |
| Full Idea: Lewis revised his analysis of duplication because he had assumed that as a matter of necessity perfectly natural properties are intrinsic, and that necessarily how a thing is intrinsically is determined completely by the natural properties it has. | |
| From: report of David Lewis (Defining 'Intrinsic' (with Rae Langton) [1998]) by Ross P. Cameron - Intrinsic and Extrinsic Properties 'Analysis' | |
| A reaction: [This compares Lewis 1986:61 with Langton and Lewis 1998] I am keen on both intrinsic and on natural properties, but I have not yet confronted this little problem. Time for a displacement activity, I think.... |
| 15458 | A property is 'intrinsic' iff it can never differ between duplicates [Lewis] |
| Full Idea: A property is 'intrinsic' iff it never can differ between duplicates; iff whenever two things (actual or possible) are duplicates, either both of them have the property or both of them lack it. | |
| From: David Lewis (Defining 'Intrinsic' (with Rae Langton) [1998], IV) | |
| A reaction: This leaves me wondering how one could arrive at a precise definition of 'duplicates'. Can it be done without mentioning that they have the same intrinsic properties? |
| 15459 | Ellipsoidal stars seem to have an intrinsic property which depends on other objects [Lewis] |
| Full Idea: The property of being an ellipsoidal star would seem offhand to be a basic intrinsic property, but it is incompatible (nomologically) with being an isolated object. | |
| From: David Lewis (Defining 'Intrinsic' (with Rae Langton) [1998], V) | |
| A reaction: Another nice example from Lewis. It makes you wonder whether the intrinsic/extrinsic distinction should go. Modern physics, with its 'entanglements', doesn't seem to suit the distinction. |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
| From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |