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) |
| 22329 | Logic is highly general truths abstracted from reality [Russell, by Glock] |
| Full Idea: In 1911 Russell held that the propositions of logic are supremely general truths about the most pervasive traits of reality, to which we have access by abstraction from non-logical propositions. | |
| From: report of Bertrand Russell (Philosophical Implications of Mathematical logic [1911]) by Hans-Johann Glock - What is Analytic Philosophy? 2.4 | |
| A reaction: Glock says the rival views were Mill's inductions, psychologism, and Frege's platonism. Wittgenstein converted Russell to a fifth view, that logic is empty tautologies. I remain resolutely attached to Russell's abstraction view. |
| 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) |
| 21569 | It is good to generalise truths as much as possible [Russell] |
| Full Idea: It is a good thing to generalise any truth as much as possible. | |
| From: Bertrand Russell (Philosophical Implications of Mathematical logic [1911], p.289) | |
| A reaction: An interesting claim, which seems to have a similar status to Ockham's Razor. Its best justification is pragmatic, and concerns strategies for coping with a big messy world. Russell's defence is in 'as much as possible'. |
| 6853 | Consequentialism wrongly assumes a clear line between an act and its consequences [Crisp,R] |
| Full Idea: What is wrong with consequentialism is that it assumes that there is agreement about when an act ends and when its consequences begin. | |
| From: Roger Crisp (Interview with Baggini and Stangroom [2001], p.31) | |
| A reaction: I certainly agree that this is a mistake in consequentialism, which has a crude idea of what an action is, though I am not convinced that this is the key fault in the theory. The theory doesn't distinguish acts by people from those by machines. |
| 6854 | Does the environment have value in itself? [Crisp,R] |
| Full Idea: The debate which dominates environmental ethics is whether the environment has value in itself. | |
| From: Roger Crisp (Interview with Baggini and Stangroom [2001], p.37) | |
| A reaction: I say the answer has to be 'yes'. If it only has value in relation to conscious or human life, it will be difficult to demonstrate that they have value. Pleasure and pain, or wisdom, can't imply value on their own; a bigger picture is needed. |