14 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) |
16776 | Substance is an intrinsic thing, so parts of substances can't also be intrinsic things [Duns Scotus] |
Full Idea: Substance ...is an ens per se. No part of a substance is an ens per se when it is part of a substance, because then it would be a particular thing, and one substance would be a particular thing from many things, which does not seem to be true. | |
From: John Duns Scotus (In Praed. [1300], 15.1), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 26.1 | |
A reaction: The tricky bit is 'when it is a part of a substance', meaning a substance must cease to be a substance when it is subsumed into some greater substance. Maybe. Drops of water? Molecules? Bricks? Cells? |
7127 | If men are good you should keep promises, but they aren't, so you needn't [Machiavelli] |
Full Idea: If all men were good, promising-breaking would not be good, but because they are bad and do not keep their promises to you, you likewise do not have to keep yours to them. | |
From: Niccolo Machiavelli (The Prince [1513], Ch.18) | |
A reaction: A rather depressing proposal to get your promise-breaking in first, based on the pessimistic view that people cannot be improved. The subsequent history of ethics in Europe showed Machiavelli to be wrong. Gentlemen began to keep their word. |
6309 | The principle foundations of all states are good laws and good armies [Machiavelli] |
Full Idea: The principle foundations of all states are good laws and good armies. | |
From: Niccolo Machiavelli (The Prince [1513], Ch.11) | |
A reaction: We may be wondering, since 1945, whether a good army is any longer essential, but it would be a foolish modern state which didn't at least form a strong alliance with a state which had a strong army. Fertile land is a huge benefit to a state. |
6305 | To retain a conquered state, wipe out the ruling family, and preserve everything else [Machiavelli] |
Full Idea: If a ruler acquires a state and is determined to keep it, he observes two cautions: he wipes out the family of their long-established princes; and he does not change either their laws or their taxes; in a short time they will unite with his old princedom. | |
From: Niccolo Machiavelli (The Prince [1513], Ch.3) | |
A reaction: This nicely illustrates the firmness of purpose for which Machiavelli has become a byword. The question is whether Machiavelli had enough empirical evidence to support this induction. The British in India seem to have been successful without it. |
6306 | People are vengeful, so be generous to them, or destroy them [Machiavelli] |
Full Idea: Men should be either treated generously or destroyed, because they take revenge for slight injuries. | |
From: Niccolo Machiavelli (The Prince [1513], Ch.3) | |
A reaction: This sounds like good advice, and works quite well in school teaching too. It seems like advice drawn from the growth of the Roman Empire, rather than from dealing with sophisticated and educated people. |
6308 | A sensible conqueror does all his harmful deeds immediately, because people soon forget [Machiavelli] |
Full Idea: A prudent conqueror makes a list of all the harmful deeds he must do, and does them all at once, so that he need not repeat them every day, which then makes men feel secure, and gains their support by treating them well. | |
From: Niccolo Machiavelli (The Prince [1513], Ch.8) | |
A reaction: This might work for a new government in a democracy, or a new boss in a business. It sounds horribly true; dreadful deeds done a long time ago can be completely forgotten, as when reformed criminals become celebrities. |
6307 | A desire to conquer, and men who do it, are always praised, or not blamed [Machiavelli] |
Full Idea: It is very natural and normal to wish to conquer, and when men do it who can, they always will be praised, or not blamed. | |
From: Niccolo Machiavelli (The Prince [1513], Ch.3) | |
A reaction: This view seems shocking to us, but it seems to me that this was a widely held view up until the time of Nietzsche, but came to a swift end with the invention of the machine gun in about 1885, followed by the heavy bomber and atomic bomb. |
7486 | Machiavelli emancipated politics from religion [Machiavelli, by Watson] |
Full Idea: Machiavelli emancipated politics from religion. | |
From: report of Niccolo Machiavelli (The Prince [1513]) by Peter Watson - Ideas Ch.24 | |
A reaction: Interestingly, he seems to have done it by saying that ideals are irrelevant to politics, but gradually secular ideals crept back in (sometimes disastrously). A balance needs to be struck on idealism. |