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) |
| 23366 | We see nature's will in the ways all people are the same [Epictetus] |
| Full Idea: The will of nature may be learned from those things in which we do not differ from one another. | |
| From: Epictetus (The Handbook [Encheiridion] [c.58], 26) | |
| A reaction: There you go! This is the rule for anthropologists on field trips. And it guides us towards a core of essential human nature. But it neglects the way that nature is expressed in different cultures, which is also important. |
| 4022 | Epictetus says we should console others for misfortune, but not be moved by pity [Epictetus, by Taylor,C] |
| Full Idea: The injunction of Epictetus is well known, that in commiserating with another for his misfortune, we ought to talk consolingly, but not be moved by pity. | |
| From: report of Epictetus (The Handbook [Encheiridion] [c.58], §16) by Charles Taylor - Sources of the Self §15.1 | |
| A reaction: This goes strongly against the grain of the Christian tradition, but strikes me as an appealing attitude (even if I am the sufferer). |
| 23365 | If someone is weeping, you should sympathise and help, but not share his suffering [Epictetus] |
| Full Idea: When you see someone weeping is sorrow …do not shrink from sympathising with him, and even groaning with him, but be careful not to groan inwardly too. | |
| From: Epictetus (The Handbook [Encheiridion] [c.58], 16) | |
| A reaction: The point is that the person's suffering is an 'indifferent' because nothing can be done about it, and we should only really care about what we are able to choose. He is not opposed to the man's suffering, or his need for support. |
| 23368 | Perhaps we should persuade culprits that their punishment is just? [Epictetus] |
| Full Idea: The governor Agrippinus would try to persuade those whom he sentenced that it was proper for them to be sentenced, …just as the physician persuades a patient to accept their treatment. | |
| From: Epictetus (The Handbook [Encheiridion] [c.58], 22) | |
| A reaction: This resembles the Contractualism of T.H. Scanlon (that actions are good if you can justify them to those involved). It may be possible to persuade people by the use of sophistry and lies. Nevertheless, a fairly civilise proposal. |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |