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) |
| 19494 | Fictionalism allows that simulated beliefs may be tracking real facts [Yablo] |
| Full Idea: The fictionalist offers the option that your simulated beliefs and assertions may be tracking a realm of genuine facts, or a realm of what you take to be facts. | |
| From: Stephen Yablo (Go Figure: a Path through Fictionalism [2001], 13) | |
| A reaction: This means that fictionalism does not have to be an error theory. That is, we aren't mistakenly believing something that we actually made up. Instead we are sensibly believing something we know to be not literally true. Love it. |
| 19493 | Governing possible worlds theory is the fiction that if something is possible, it happens in a world [Yablo] |
| Full Idea: The governing fiction of possible worlds theory says that whenever something is possible, there is a world where it happens. | |
| From: Stephen Yablo (Go Figure: a Path through Fictionalism [2001], 05) | |
| A reaction: This sounds like the only sensible attitude to possible worlds I can think of. |
| 3364 | All mental phenomena contain an object [Brentano] |
| Full Idea: Every mental phenomenon contains something as object within itself. | |
| From: Franz Brentano (Psychology from an empirical standpoint [1874], p. 88), quoted by Jaegwon Kim - Philosophy of Mind p.21 | |
| A reaction: This gives rise to the slogan that 'intentionality is the mark of the mental', which notoriously seems to miss out the phenomenal aspect of mental life. We note now, though, that even emotions have objects. |
| 3225 | Mental unity suggests that qualia and intentionality must connect [Brentano, by Rey] |
| Full Idea: Brentano's thesis is that all mental phenomena are intentional i.e. representational. Support for this view is that assimilating phenomenal experience to attitudes we explain the essential unity of the mind. | |
| From: report of Franz Brentano (Psychology from an empirical standpoint [1874]) by Georges Rey - Contemporary Philosophy of Mind 11.5 | |
| A reaction: Unifying intentionality and qualia in a single theory looks like a good move, but which one has priority? Evolutionary theory says priority goes to whatever produces behaviour. My intuition is that qualia are more basic - in tiny insects, say. |