12 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) |
| 22392 | Morality is inescapable, in descriptive words such as 'dishonest', 'unjust' and 'uncharitable' [Foot] |
| Full Idea: There is a sense in which morality is inescapable - in moral epithets such as 'dishonest', 'unjust', 'uncharitable'; these do not cease to apply to a man because he is indifferent to the ends of morality: they may indeed apply because of his indifference. | |
| From: Philippa Foot (Morality as system of hypothetical imperatives [1972], p.172 n15) | |
| A reaction: Odysseus was admired for lying, and charity wasn't a virtue in the ancient world. They won't go away as factual descriptions, but the values attached to them vary quite a lot. |
| 23685 | Reason is not a motivator of morality [Foot, by Hacker-Wright] |
| Full Idea: In her middle period she changed her mind, and attacks moral rationalism. | |
| From: report of Philippa Foot (Morality as system of hypothetical imperatives [1972]) by John Hacker-Wright - Philippa Foot's Moral Thought Intro | |
| A reaction: That is, she doubted whether moral reasons are sufficient to motivate moral actions, which presumably therefore need desires, as the Humeans claimed. Reasons rely on merely hypothetical rules. |
| 23691 | Rejecting moral rules may be villainous, but it isn't inconsistent [Foot] |
| Full Idea: The man who rejects morality because he sees no reason to obey its rules can be convicted of villainy but not of inconsistency. | |
| From: Philippa Foot (Morality as system of hypothetical imperatives [1972], p.161) | |
| A reaction: This is 'middle period' Foot, when she decided that Hume was right about the need for a desire as moral motivator. Before and after this time, she thought there were reasons to be moral, as well as desires. |
| 16764 | The soul conserves the body, as we see by its dissolution when the soul leaves [Toletus] |
| Full Idea: Every accident of a living thing, as well as all its organs and temperaments and its dispositions are conserved by the soul. We see this from experience, since when that soul recedes, all these dissolve and become corrupted. | |
| From: Franciscus Toletus (Commentary on 'De Anima' [1572], II.1.1), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 24.5 | |
| A reaction: A nice example of observing a phenemonon, but not being able to observe the dependence relation the right way round. Compare Descartes in Idea 16763. |
| 22391 | Saying we 'ought to be moral' makes no sense, unless it relates to some other system [Foot] |
| Full Idea: 'One ought to be moral' makes no sense at all unless the 'ought' has the moral subscript, giving a tautology, or else relates morality to some other system such as prudence or etiquette. | |
| From: Philippa Foot (Morality as system of hypothetical imperatives [1972], p.169 n18) | |
| A reaction: This aims to undercut the Kantian view that morality is an absolute call to duty (filling us with wonder, like the starry heavens). Foot aims to root morality in the real world. |
| 22389 | Morality no more consists of categorical imperatives than etiquette does [Foot] |
| Full Idea: Moral judgements have no better claim to be categorical imperatives than do statements about matters of etiquette. | |
| From: Philippa Foot (Morality as system of hypothetical imperatives [1972], p.164) | |
| A reaction: Her claim is that all moral judgements are responses to situations, and so are hypothetical. This judgement of hers is the culmination of a careful discussion. |