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) |
| 6316 | We translate in a way that makes the largest possible number of statements true [Wilson,NL] |
| Full Idea: We select as designatum that individual which will make the largest possible number of statements true. | |
| From: N.L. Wilson (Substances without Substrata [1959]), quoted by Willard Quine - Word and Object II.§13 n | |
| A reaction: From the Quine's reference, it sounds as if Wilson was the originator of the well-known principle of charity, later taken up by Davidson. If so, he should be famous, because it strikes me as a piece of fundamental and important wisdom. |
| 21375 | Reason can be vicious, and great crimes have to be rational [Schopenhauer] |
| Full Idea: Reasonable and vicious are quite consistent with each other, in fact, only through their union are great and far-reaching crimes possible. | |
| From: Arthur Schopenhauer (On the Basis of Morality [1841], p.83), quoted by Christopher Janaway - Schopenhauer 7 'Against' | |
| A reaction: This is opposed to Kant, who always looks wildly optimistic in his hope that high rationality entails a morally good will. Good people seem to have a fairly irrational empathy with their fellow citizens. |
| 21379 | Man's three basic ethical incentives are egoism, malice and compassion [Schopenhauer] |
| Full Idea: Man's three fundamental ethical incentives, egoism, malice and compassion, are present in everyone in different and incredibly unequal proportions. In accordance with them, motives will operate on man and actions will ensue. | |
| From: Arthur Schopenhauer (On the Basis of Morality [1841], p.192), quoted by Christopher Janaway - Schopenhauer 7 'Egoism' | |
| A reaction: A well chosen trio. Kant would be shocked that he has left out duty, which is supposed to rise above such feelings. |
| 21376 | Philosophy treats animals as exploitable things, ignoring the significance of their lives [Schopenhauer] |
| Full Idea: In philosophical morals animals are mere 'things', mere means to any end whatsoever. ...Shame on such a morality, that fails to recognise the eternal essence that lives in every living thing, and shines forth with inscrutable significance from all eyes. | |
| From: Arthur Schopenhauer (On the Basis of Morality [1841], p.96), quoted by Christopher Janaway - Schopenhauer 7 'Against' | |
| A reaction: Good. I find Kant's theoretical indifference to animals very creepy (despite his kind attitude to them). And I also think the utilitarians are wrong to only value animals for their pain, as if any animal could be shredded for fun, if it felt no pain. |