Combining Texts

All the ideas for 'On the Question of Absolute Undecidability', 'The Ages of the World' and 'Properties'

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11 ideas

4. Formal Logic / F. Set Theory ST / 1. Set Theory
Mathematical set theory has many plausible stopping points, such as finitism, and predicativism [Koellner]
     Full Idea: There are many coherent stopping points in the hierarchy of increasingly strong mathematical systems, starting with strict finitism, and moving up through predicativism to the higher reaches of set theory.
     From: Peter Koellner (On the Question of Absolute Undecidability [2006], Intro)
'Reflection principles' say the whole truth about sets can't be captured [Koellner]
     Full Idea: Roughly speaking, 'reflection principles' assert that anything true in V [the set hierarchy] falls short of characterising V in that it is true within some earlier level.
     From: Peter Koellner (On the Question of Absolute Undecidability [2006], 2.1)
5. Theory of Logic / K. Features of Logics / 5. Incompleteness
We have no argument to show a statement is absolutely undecidable [Koellner]
     Full Idea: There is at present no solid argument to the effect that a given statement is absolutely undecidable.
     From: Peter Koellner (On the Question of Absolute Undecidability [2006], 5.3)
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
There are at least eleven types of large cardinal, of increasing logical strength [Koellner]
     Full Idea: Some of the standard large cardinals (in order of increasing (logical) strength) are: inaccessible, Mahlo, weakly compact, indescribable, Erdös, measurable, strong, Wodin, supercompact, huge etc. (...and ineffable).
     From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4)
     A reaction: [I don't understand how cardinals can have 'logical strength', but I pass it on anyway]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner]
     Full Idea: To the extent that we are justified in accepting Peano Arithmetic we are justified in accepting its consistency, and so we know how to expand the axiom system so as to overcome the limitation [of Gödel's Second Theorem].
     From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.1)
     A reaction: Each expansion brings a limitation, but then you can expand again.
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
Arithmetical undecidability is always settled at the next stage up [Koellner]
     Full Idea: The arithmetical instances of undecidability that arise at one stage of the hierarchy are settled at the next.
     From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4)
8. Modes of Existence / B. Properties / 2. Need for Properties
We accept properties because of type/tokens, reference, and quantification [Edwards]
     Full Idea: Three main reasons for thinking properties exist: the one-over-many argument (that a type can have many tokens), the reference argument (to understand predicates and singular terms), and the quantification argument (that we quantify over them).
     From: Douglas Edwards (Properties [2014], 1.1)
     A reaction: [Bits in brackets are compressions of his explanations]. I don't find any of these remotely persuasive. Why would we infer how the world is, simply from how we talk about or reason about the world? His first reason is the only interesting one.
8. Modes of Existence / B. Properties / 10. Properties as Predicates
Quineans say that predication is primitive and inexplicable [Edwards]
     Full Idea: The Quinean claims that the application of a predicate cannot, in principle, be explained - it is a 'primitive' fact.
     From: Douglas Edwards (Properties [2014], 4.4)
     A reaction: I am not clear what 'principle' could endorse this claim. There just seems to be a possible failure of all the usual attempts at explaining predication.
8. Modes of Existence / E. Nominalism / 2. Resemblance Nominalism
Resemblance nominalism requires a second entity to explain 'the rose is crimson' [Edwards]
     Full Idea: For resemblance nominalism the sentence 'the rose is crimson' commits us to at least one other entity that the rose resembles in order for it to be crimson.
     From: Douglas Edwards (Properties [2014], 5.5.2)
     A reaction: If the theory really needs this, then it has just sunk without trace. It can't suddenly cease to be crimson when the last resembling entity disappears.
9. Objects / C. Structure of Objects / 8. Parts of Objects / c. Wholes from parts
That a whole is prior to its parts ('priority monism') is a view gaining in support [Edwards]
     Full Idea: The view of 'priority monism' - that the whole is prior to its parts - is controversial, but has been growing in support
     From: Douglas Edwards (Properties [2014], 5.4.4)
     A reaction: The simple and plausible thought is, I take it, that parts only count as parts when a whole comes into existence, so a whole is needed to generate parts. Thus the whole must be prior to the parts. Fine by me.
23. Ethics / C. Virtue Theory / 2. Elements of Virtue Theory / e. Character
We don't choose our characters, yet we still claim credit for the actions our characters perform [Schelling]
     Full Idea: Nobody has chosen their character; and yet this does not stop anybody attributing the action which follows from his character to themself as a free action.
     From: Friedrich Schelling (The Ages of the World [1810], I.93)
     A reaction: This pinpoints a very nice ambivalence about our attitudes to our own characters. We all have some pride and shame about who we are, without having chosed who we are. At least when we are young. But we make the bed we lie in.