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) |
3291 | Emergent properties appear at high levels of complexity, but aren't explainable by the lower levels [Nagel] |
Full Idea: The supposition that a diamond or organism should truly have emergent properties is that they appear at certain complex levels of organisation, but are not explainable (even in principle) in terms of any more fundamental properties of the system. | |
From: Thomas Nagel (Panpsychism [1979], p.186) |
7260 | If there are intuited moral facts, why should we care about them? [Dancy,J] |
Full Idea: Critics asked (of intuitionism) why, if moral facts are as the intuitionists say, we should care about them at all. | |
From: Jonathan Dancy (Intuitionism [1991]) | |
A reaction: It is a good question, as we don't care much about other a priori truths, such as the square root of 169. |
7261 | Internalists say that moral intuitions are motivating; externalist say a desire is also needed [Dancy,J] |
Full Idea: There is an internalist view of intuitionism, saying that to accept that one's action is wrong is itself to be motivated not to do it. Externalists (like Ross) say that moral judgements need the help of an independent desire to motivate us. | |
From: Jonathan Dancy (Intuitionism [1991]) | |
A reaction: The internalists would be closer to Kant or Plato (for whom reason or pure ideas motivate), while externalist would favour Hume's belief/desire account of human actions. I like Kant and Plato, but Hume is more plausible. Dancy disagrees (Idea 7262). |
7262 | Obviously judging an action as wrong gives us a reason not to do it [Dancy,J] |
Full Idea: It is ludicrous to say that we might accept an action is outrageously wrong and still think of this as not in itself giving us good reason to hold back. | |
From: Jonathan Dancy (Intuitionism [1991]) | |
A reaction: If we think of some dreadful man-made famine in a remote continent, our judgement may well give a reason to act, but apathy usually intervenes. We are discussing a purely theoretical motive on the one hand, and an actual motivator on the other. |
7265 | Moral facts are not perceived facts, but perceived reasons for judgements [Dancy,J] |
Full Idea: I intend to suggest that moral facts are best thought of not as facts perceived but as reasons recognised in the exercise of practical moral judgement. | |
From: Jonathan Dancy (Intuitionism [1991]) | |
A reaction: I'm not convinced by this modified version. Why should the fact that someone is in pain be, in itself, a reason to prevent it? There are different cultural traditions for response to the pain of others. We are the squeamish tradition. |
3290 | Given the nature of heat and of water, it is literally impossible for water not to boil at the right heat [Nagel] |
Full Idea: Given what heat is and what water is, it is literally impossible for water to be heated beyond a certain point at normal atmospheric pressure without boiling. | |
From: Thomas Nagel (Panpsychism [1979], p.186) |