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) |
| 19594 | General statements about nature are not valid [Novalis] |
| Full Idea: General statements are not valid in the study of nature. | |
| From: Novalis (Last Fragments [1800], 17) | |
| A reaction: This is his striking obsession with the particularity and fine detail of nature. Alexander von Humbolt was exploring nature in S.America in this year. It sounds wrong about physics, but possibly right about biology. |
| 19596 | The whole body is involved in the formation of thoughts [Novalis] |
| Full Idea: In the formation of thoughts all parts of the body seem to me to be working together. | |
| From: Novalis (Last Fragments [1800], 20) | |
| A reaction: I can only think that Spinoza must be behind this thought, or La Mettrie. It seems a strikingly unusual intuition for its time, when almost everyone takes a spiritual sort of dualism for granted. |
| 7810 | The 'Eumenides' of Aeschylus shows blood feuds replaced by law [Aeschylus, by Grayling] |
| Full Idea: The 'Eumenides' of Aeschylus tells how the old rule of revenge and blood feud was replaced by a due process of law before a civil jury. | |
| From: report of Aeschylus (The Eumenides [c.458 BCE]) by A.C. Grayling - What is Good? Ch.2 | |
| A reaction: Compare Idea 1659, where this revolution is attributed to Protagoras (a little later than Aeschylus). I take the rule of law and of society to be above all the rule of reason, because the aim is calm objectivity instead of emotion. |
| 19593 | Persons are shaped by a life history; splendid persons are shaped by world history [Novalis] |
| Full Idea: What is it that shapes a person if not his life history? And in the same way a splendid person is shaped by nothing other than world history. Many people live better in the past and in the future than in the present. | |
| From: Novalis (Last Fragments [1800], 15) | |
| A reaction: Clearly there is a lot to be said for splendid people who live entirely in the present (such as jazz musicians). Some people do have an awesomely wide historical perspective on their immediate lives. Palaeontology is not the master discipline though! |
| 19595 | Nature is a whole, and its individual parts cannot be wholly understood [Novalis] |
| Full Idea: Nature is a whole - in which each part in itself can never be wholly understood. | |
| From: Novalis (Last Fragments [1800], 18) | |
| A reaction: This doesn't seem right when studying some item in a laboratory, but it seems undeniable when you consider the history and future of each item. |
| 19592 | The basic relations of nature are musical [Novalis] |
| Full Idea: Musical relations seem to me to be actually the basic relations of nature. | |
| From: Novalis (Last Fragments [1800], 10) | |
| A reaction: Novalis shows no signs of being a pythagorean, and then suddenly comes out with this. I suppose if you love music, this thought should float into your mind at regular intervals, because the power of music is so strong. Does he mean ratios? |