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All the ideas for 'fragments/reports', 'Anti-essentialism' and 'Review of Chihara 'Struct. Accnt of Maths''

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13 ideas

5. Theory of Logic / C. Ontology of Logic / 1. Ontology of Logic
Logical space is abstracted from the actual world [Stalnaker]
     Full Idea: Logical space is not given independently of the individuals that occupy it, but is abstracted from the world as we find it.
     From: Robert C. Stalnaker (Anti-essentialism [1979], p.85)
     A reaction: I very much like the second half of this idea, and am delighted to find Stalnaker endorsing it. I take the logical connectives to be descriptions of how things behave, at a high level of generality.
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
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.
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
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.
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'?
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
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.
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.
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).
9. Objects / C. Structure of Objects / 7. Substratum
For the bare particular view, properties must be features, not just groups of objects [Stalnaker]
     Full Idea: If we are to make sense of the bare particular theory, a property must be not just a rule for grouping individuals, but a feature of individuals in virtue of which they may be grouped.
     From: Robert C. Stalnaker (Anti-essentialism [1979], p.76)
     A reaction: He is offering an objection to the thoroughly extensional account of properties that is found in standard possible worlds semantics. Quite right too. We can't give up on the common sense notion of a property.
9. Objects / D. Essence of Objects / 7. Essence and Necessity / a. Essence as necessary properties
An essential property is one had in all the possible worlds where a thing exists [Stalnaker]
     Full Idea: If necessity is explained in terms of possible worlds, ...then an essential property is a property that a thing has in all possible worlds in which it exists.
     From: Robert C. Stalnaker (Anti-essentialism [1979], p.71)
     A reaction: This seems to me to be a quite shocking confusion of necessary properties with essential properties. The point is that utterly trivial properties can be necessary, but in no way part of the real essence of something.
9. Objects / D. Essence of Objects / 7. Essence and Necessity / b. Essence not necessities
Necessarily self-identical, or being what it is, or its world-indexed properties, aren't essential [Stalnaker]
     Full Idea: We can remain anti-essentialist while allowing some necessary properties: those essential to everything (self-identity), relational properties (being what it is), and world-indexed properties (being snub-nosed-only-in-Kronos).
     From: Robert C. Stalnaker (Anti-essentialism [1979], p.73)
     A reaction: [a summary] He defined essential properties as necessary properties (Idea 12761), and now backpeddles. World-indexed properties are an invention of Plantinga, as essential properties to don't limit individuals. But they are necessary, not essential!
9. Objects / D. Essence of Objects / 15. Against Essentialism
Bare particular anti-essentialism makes no sense within modal logic semantics [Stalnaker]
     Full Idea: I argue that one cannot make semantical sense out of bare particular anti-essentialism within the framework of standard semantics for modal logic.
     From: Robert C. Stalnaker (Anti-essentialism [1979], p.71)
     A reaction: Stalnaker characterises the bare particular view as ANTI-essentialist, because he has defined essence in terms of necessary properties. The bare particular seems to allow the possibility of Aristotle being a poached egg.
10. Modality / E. Possible worlds / 3. Transworld Objects / a. Transworld identity
Why imagine that Babe Ruth might be a billiard ball; nothing useful could be said about the ball [Stalnaker]
     Full Idea: I cannot think of any point in making the counterfactual supposition that Babe Ruth is a billiard ball; there is nothing I can say about him in that imagined state that I could not just as well say about billiard balls that are not him.
     From: Robert C. Stalnaker (Anti-essentialism [1979], p.79)
     A reaction: A bizarrely circumspect semanticists way of saying that Ruth couldn't possibly be a billiard ball! Would he say the same about a group of old men in wheelchairs, one of whom IS Babe Ruth?
21. Aesthetics / C. Artistic Issues / 7. Art and Morality
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?