Combining Texts

All the ideas for 'Mahaprajnaparamitashastra', 'Review of Chihara 'Struct. Accnt of Maths'' and 'Distinct Indiscernibles and the Bundle Theory'

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10 ideas

6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory

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.

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism

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'?

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique

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).

8. Modes of Existence / D. Universals / 3. Instantiated Universals

10197	An immanent universal is wholly present in more than one place [Zimmerman,DW]
	Full Idea: An immanent universal will routinely be 'at some distance from itself', in the sense that it is wholly present in more than one place.
	From: Dean W. Zimmerman (Distinct Indiscernibles and the Bundle Theory [1997], p.306)
	A reaction: This is the Aristotelian view, which sounds distinctly implausible in this formulation. Though I suppose redness is wholly present in a tomato, in the way that fourness is wholly present in the Horsemen of the Apocalypse. How many rednesses are there?

9. Objects / F. Identity among Objects / 7. Indiscernible Objects

10198	If only two indiscernible electrons exist, future differences must still be possible [Zimmerman,DW]
	Full Idea: If nothing existed except two electrons, which are indiscernible, it remains possible that differences will emerge later. Even if this universe has eternal symmetry, such differences are still logically, metaphysically, physically and causally possible.
	From: Dean W. Zimmerman (Distinct Indiscernibles and the Bundle Theory [1997], p.306)
	A reaction: The question then is whether the two electrons have hidden properties that make differences possible. Zimmerman assumes that 'laws' of an indeterministic kind will do the job. I doubt that. Can differences be discerned after the event?

10199	Discernible differences at different times may just be in counterparts [Zimmerman,DW]
	Full Idea: Possible differences which may later become discernible could be treated as differences in a counterpart, which is similar to, but not identical with, the original object.
	From: Dean W. Zimmerman (Distinct Indiscernibles and the Bundle Theory [1997], p.307)
	A reaction: [compressed] This is a reply to Idea 10198, which implies that two things could never be indiscernible over time, because of their different possibilities. One must then decide issues about rigid designation and counterparts.

23. Ethics / C. Virtue Theory / 3. Virtues / a. Virtues

7903	The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna]
	Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom.
	From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88)
	A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate').