10 ideas
| 13750 | Analysis aims at the structure of facts, which are needed to give a rationale to analysis [Urmson, by Schaffer,J] |
| Full Idea: Urmson explains the direction of analysis as 'towards a structure...more nearly similar to the structure of the fact', adding that this metaphysical picture is needed as a 'rationale of the practice of analysis'. | |
| From: report of J.O. Urmson (Philosophical Analysis [1956], p.24-5) by Jonathan Schaffer - On What Grounds What n30 | |
| A reaction: In other words, only realists can be truly motivated to keep going with analysis. Merely analysing language-games is doable, but hardly exciting. |
| 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) |
| 1471 | It may be hard to verify that we have become immortal, but we could still then verify religious claims [Hick, by PG] |
| Full Idea: Verification of religious claims after death is only possible if the concept of surviving death is intelligible, and we can understand the concept of immortality, despite difficulties in being certain that we had reached it. | |
| From: report of John Hick (Theology and Verification [1960], IV) by PG - Db (ideas) |
| 1470 | Belief in an afterlife may be unverifiable in this life, but it will be verifiable after death [Hick, by PG] |
| Full Idea: Religion is capable of 'eschatological verification', by reaching evidence at the end of life, even though falsification of its claims is never found in this life; a prediction of coming to a Celestial City must await the end of the journey. | |
| From: report of John Hick (Theology and Verification [1960], III) by PG - Db (ideas) |
| 1469 | Some things (e.g. a section of the expansion of PI) can be verified but not falsified [Hick, by PG] |
| Full Idea: Falsification and verification are not logically equivalent. For example, you might verify the claim that there will be three consecutive sevens in the infinite expansion of PI, but you could never falsify such a claim. | |
| From: report of John Hick (Theology and Verification [1960], §II) by PG - Db (ideas) |