10 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) |
| 23283 | Necessity implies possibility, but in experience it matters which comes first [Williams,B] |
| Full Idea: Any notion of necessity must carry with it a corresponding notion of impossibility, …but it can make a difference which one of them presents itself first and more naturally. | |
| From: Bernard Williams (Practical Necessity [1982], p.127) | |
| A reaction: I like this because it connects modality with experience, rather than with formal logic. It seems right that in life we immediately see either a necessity or an impossibility, and inferring the other case is an afterthought. |
| 20327 | Modern attention has moved from the intrinsic properties of art to its relational properties [Lamarque/Olson] |
| Full Idea: In modern discussions, rather than look for intrinsic properties of objects, including aesthetic or formal properties, attention has turned to extrinsic or relational properties, notably of a social, historical, or 'institutional' nature. | |
| From: Lamargue,P/Olson,SH (Introductions to 'Aesthetics and the Phil of Art' [2004], Pt 1) | |
| A reaction: Lots of modern branches of philosophy have made this move, which seems to me like a defeat. We want to know why things have the relations they do. Just mapping the relations is superficial Humeanism. |
| 20326 | Early 20th cent attempts at defining art focused on significant form, intuition, expression, unity [Lamarque/Olson] |
| Full Idea: In the early twentieth century there were numerous attempts at defining the essence art. Significant form, intuition, the expression of emotion, organic unity, and other notions, were offered to this end. | |
| From: Lamargue,P/Olson,SH (Introductions to 'Aesthetics and the Phil of Art' [2004], Pt 1) | |
| A reaction: As far as I can see the whole of aesthetics was demolished in one blow by Marcel Duchamp's urinal. Artists announce: we will tell you what art is; you should just sit and listen. Compare the invention of an anarchic sport. |
| 20330 | The dualistic view says works of art are either abstract objects (types), or physical objects [Lamarque/Olson] |
| Full Idea: The dualistic view of the arts holds that works of art come in two fundamentally different kinds: those that are abstract entities, i.e. types, and those that are physical objects (tokens). | |
| From: Lamargue,P/Olson,SH (Introductions to 'Aesthetics and the Phil of Art' [2004], Pt 2) | |
| A reaction: Paintings are the main reason for retaining physical objects. Strawson 1974 argues that paintings are only physical because we cannot yet perfectly reproduce them. I agree. Works of art are types, not tokens. |