15 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) |
| 16755 | The possible Aristotelian view that forms are real and active principles is clearly wrong [Fine,K, by Pasnau] |
| Full Idea: Aristotle seems to have a possible basis for the belief [in individual forms], namely that forms are real and active principles in the world, which is denied by any right-minded modern. | |
| From: report of Kit Fine (A Puzzle Concerning Matter and Form [1994], p.19) by Robert Pasnau - Metaphysical Themes 1274-1671 24.3 n8 | |
| A reaction: Pasnau says this is the view of forms promoted by the scholastics, whereas Aristotle's own view should be understood as 'metaphysical'. |
| 21332 | We don't get a love of 'order' from nature - which is thoroughly chaotic [Mill] |
| Full Idea: Even the love of 'order' which is thought to be a following of the ways of nature is in fact a contradiction of them. All which people are accustomed to deprecate as 'disorder' is precisely a counterpart of nature's ways. | |
| From: John Stuart Mill (Nature and Utility of Religion [1874], p.116) | |
| A reaction: The Greeks elevated the idea that the cosmos was orderly, but almost entirely based on the regular movement of the planets. They turned a blind eye to the messy bits of nature. As you magnify nature, order and chaos seem to alternate. |
| 21333 | Evil comes from good just as often as good comes from evil [Mill] |
| Full Idea: If good frequently comes out of evil, the converse fact, evil coming out of good, is equally common. | |
| From: John Stuart Mill (Nature and Utility of Religion [1874], p.117) | |
| A reaction: Mill surmises that on the whole good comes from good, and evil from evil, but the point is that the evidence doesn't favour the production of increased good. |
| 21335 | Belief that an afterlife is required for justice is an admission that this life is very unjust [Mill] |
| Full Idea: The necessity of redressing the balance [of injustice] is deemed one of the strongest arguments for another life after death, which amounts to an admission that the order of things in this life is often an example of injustice, not justice. | |
| From: John Stuart Mill (Nature and Utility of Religion [1874]) | |
| A reaction: It certainly seems that an omnipotent God could administer swift justice in this life. If the whole point is that we need freedom of will, then why is justice administered at a much later date? The freedom seems to be illusory. |
| 21334 | No necessity ties an omnipotent Creator, so he evidently wills human misery [Mill] |
| Full Idea: If a Creator is assumed to be omnipotent, if he bends to a supposed necessity, he himself makes the necessity which he bends to. If the maker of the world can all that he will, he wills misery, and there is no escape from the conclusion. | |
| From: John Stuart Mill (Nature and Utility of Religion [1874], p.119) | |
| A reaction: If you add that the Creator is supposed to be perfectly benevolent, you arrive at the paradox which Mackie spells out. Is the correct conclusion that God exists, and is malevolent? Mill doesn't take that option seriously. |
| 21329 | Nature dispenses cruelty with no concern for either mercy or justice [Mill] |
| Full Idea: All of this [cruel killing] nature does with the most supercilious disregard both of mercy and of justice, emptying her shafts upon the best and noblest indifferently with the meanest and worst | |
| From: John Stuart Mill (Nature and Utility of Religion [1874], p.115) | |
| A reaction: The existence of an afterlife at least offers an opportunity to rectify any injustice, but that hardly meets the question of why there was injustice in the first place. It would be odd if it actually is justice, but none of us can see why that is so. |
| 21328 | Killing is a human crime, but nature kills everyone, and often with great tortures [Mill] |
| Full Idea: Killing, the most criminal act recognised by human laws, nature does once to every being that lives, and frequently after protracted tortures such as the greatest know monsters purposely inflicted on their living fellow creatures | |
| From: John Stuart Mill (Nature and Utility of Religion [1874], p.115) | |
| A reaction: We certainly don't condemn lions for savaging gazelles, but the concept of a supreme mind controlling nature forces the question. Theology needs consistency between human and divine morality, and the supposed derivation of the former from the latter. |
| 21330 | Nature makes childbirth a miserable experience, often leading to the death of the mother [Mill] |
| Full Idea: In the clumsy provision which nature has made for the perpetual renewal of animal life, ...no human being ever comes into the world but another human being is literally stretched on the rack for hours or day, not unfrequently issuing in death. | |
| From: John Stuart Mill (Nature and Utility of Religion [1874], p.116) | |
| A reaction: This is a very powerful example, which is rarely cited in modern discussions. |
| 21331 | Hurricanes, locusts, floods and blight can starve a million people to death [Mill] |
| Full Idea: Nature often takes the means by which we live. A single hurricane, a flight of locusts, or an inundation, or a trifling chemical change in an edible root, starve a million people. | |
| From: John Stuart Mill (Nature and Utility of Religion [1874], p.116) | |
| A reaction: [second sentence compressed] The 'edible root' is an obvious reference to the Irish potato famine. Some desertification had human causes, but these are telling examples. |