11 ideas
| 17884 | 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) |
| 17893 | '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) |
| 17894 | 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) |
| 17890 | 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] |
| 17887 | 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. |
| 17891 | 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) |
| 22371 | Determinism threatens free will if actions can be causally traced to external factors [Foot] |
| Full Idea: The determinism which worries the defender of free will is that if human action is subject to a universal law of causation, there will be for any action a set of sufficient conditions which can be traced back to factors outside the control of the agent. | |
| From: Philippa Foot (Free Will as Involving Determinism [1957], p.63) | |
| A reaction: She draws on Russell for this, but neither of them mention whether the causation is physical. Free will seems to imply non-physical causation. |
| 22372 | Not all actions need motives, but it is irrational to perform troublesome actions with no motive [Foot] |
| Full Idea: We do not expect that everything a rational man does should be done with a motive, ...but we do expect a man to have a motive for many things that he does, and would count anyone who constantly performed troublesome actions without a motive as irrational. | |
| From: Philippa Foot (Free Will as Involving Determinism [1957], p.66) | |
| A reaction: Interestng, because the assessment of whether someone is 'rational' therefore needs a criterion for when a motive seems required and when not. 'Significant' actions need a motive? |
| 12165 | Romantics say music expresses ideas, or the Will, or intuitions, or feelings [Scruton] |
| Full Idea: According to the Romantic theory music was an expression of something, of an idea (Hegel), of the Will (Schopenhauer), of 'intuitions' (Croce), or of feelings (Collingwood). | |
| From: Roger Scruton (The Nature of Musical Expression [1981], p.54) | |
| A reaction: Deryck Cooke was the culmination of music as expression of feeling, and Stravinsky was the greatest rebel against the whole idea of expression in music. You can set out to create interesting music which does or does not grab the emotions. |
| 12164 | Expressing melancholy is a good thing, but arousing it is a bad thing [Scruton] |
| Full Idea: To describe a piece of music as expressive of melancholy is to give a reason for listening to it; to describe it as arousing or evoking melancholy is to give a reason for avoiding it. | |
| From: Roger Scruton (The Nature of Musical Expression [1981], p.49) | |
| A reaction: Expressing sexual desire, while avoiding arousing it, is the nice challenge for a particular type of art. Would Scruton say that expressing joy is a good thing, but arousing it is bad? It is a nice observation, though. |
| 22373 | People can act out of vanity without being vain, or even vain about this kind of thing [Foot] |
| Full Idea: It makes sense to say that a man acts out of vanity on a particular occasion although he is not in general vain, or even vain about this kind of thing. | |
| From: Philippa Foot (Free Will as Involving Determinism [1957], p.69) | |
| A reaction: Aristotle tells us that virtues and vices are habits, and also have an intellectual component, implying that the person believes in that sort of behaviour. Anyone can have 'a little moment of vanity'. |