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) |
| 16756 | Substantial forms must exist, to explain the stability of metals like silver and tin [Albertus Magnus] |
| Full Idea: There is no reason why the matter in any natural thing should be stable in its nature, if it is not completed by a substantial form. But we see that silver is stable, and tin and other metals. Therefore they will seem to be perfected by substantial forms. | |
| From: Albertus Magnus (On Minerals [1260], III.1.7), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 24.2 | |
| A reaction: Illuminating. This may be the best reason for proposing substantial forms. Once materialism arrives, the so-called 'laws' of nature have to be imposed on the material to do the job - but what the hell is a law supposed to be? |
| 16718 | Primary qualities are the cause of all the other sensible qualities [Albertus Magnus] |
| Full Idea: The primary qualities of tangible things are the cause of all the other sensible qualities. | |
| From: Albertus Magnus (On 'Generation and Corruption' [1261], II.1.1), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 21.2 | |
| A reaction: This makes the primary qualities sound suspiciously like the essence. |
| 5845 | Niceratus learnt the whole of Homer by heart, as a guide to goodness [Xenophon] |
| Full Idea: Niceratus said that his father, because he was concerned to make him a good man, made him learn the whole works of Homer, and he could still repeat by heart the entire 'Iliad' and 'Odyssey'. | |
| From: Xenophon (Symposium [c.391 BCE], 3.5) | |
| A reaction: This clearly shows the status which Homer had in the teaching of morality in the time of Socrates, and it is precisely this acceptance of authority which he was challenging, in his attempts to analyse the true basis of virtue |
| 5833 | Education is the greatest of human goods [Xenophon] |
| Full Idea: Education is the greatest of human goods. | |
| From: Xenophon (Apology of Socrates [c.392 BCE], 22) | |
| A reaction: Of course, one might ask what education is for, and arrive at a greater good. If you ask what is the greatest good which a society can provide for you, or which you can give to your children, this seems to me a good answer. |