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) |
| 9089 | Knowledge is a quality existing subjectively in the soul [William of Ockham] |
| Full Idea: Knowledge is a certain quality which exists in the soul as its subject ('existens subiective in anima'). | |
| From: William of Ockham (Expositio super viii libros [1340], Prologue) | |
| A reaction: One might say here that knowledge is a property, and so it might not be susceptible to further analysis. It invites the question of how you could know by introspection that you have got it, which would be an extreme internalist view. |
| 9091 | Sometimes 'knowledge' just concerns the conclusion, sometimes the whole demonstration [William of Ockham] |
| Full Idea: Sometimes 'knowledge' means evident cognition of the conclusion alone, sometimes of the demonstration as a whole. | |
| From: William of Ockham (Expositio super viii libros [1340], Prologue) | |
| A reaction: 'Demonstration' will be something like Greek 'logos' - full understanding, ability to explain and give reasons. William is certainly right about normal usage. I know the answer in a quiz, without any requirement for justifications. |
| 9090 | Knowledge is certain cognition of something that is true [William of Ockham] |
| Full Idea: Knowledge is certain cognition of something that is true. | |
| From: William of Ockham (Expositio super viii libros [1340], Prologue) | |
| A reaction: This view has problems. William is not facing up to the sceptical questions which can shake any degree of certainty, and also that someone who lacked self-confidence might know many things while always feeling uncertain about them. 'Cognition' must go! |
| 8724 | The meaning of 'know' does not change from courtroom to living room [Unger] |
| Full Idea: There is no reason to suppose that the meaning of 'know' changes from the courtroom to the living room and back again; no more than for supposing that 'vacuum' changes from the laboratory to the cannery. | |
| From: Peter Unger (Ignorance: a Case for Scepticism [1975], 2.1) | |
| A reaction: I disagree. Lots of words change their meaning (or reference) according to context. Flat, fast, tall, clever. She 'knows a lot' certainly requires a context. The bar of justification goes up and down, and 'knowledge' changes accordingly. |
| 8722 | No one knows anything, and no one is ever justified or reasonable [Unger] |
| Full Idea: I argue for the thesis that no one ever knows about anything, ...and that consequently no one is ever justified or at all reasonable in anything. | |
| From: Peter Unger (Ignorance: a Case for Scepticism [1975], Intro) | |
| A reaction: The premiss of his book seems to be that knowledge is assumed to require certainty, and is therefore impossible. Unger has helped push us to a more relaxed and fallibilist attitude to knowledge. 'No one is reasonable' is daft! |
| 8723 | An evil scientist may give you a momentary life, with totally false memories [Unger] |
| Full Idea: The evil scientist might not only be deceiving you with his electrodes; maybe he has just created you with your ostensible memory beliefs and experiences, and for good measure he will immediately destroy you, so in the next moment you no longer exist. | |
| From: Peter Unger (Ignorance: a Case for Scepticism [1975], 1.12) | |
| A reaction: This is based on Russell's scepticism about memory (Idea 2792). Even this very train of thought may not exist, if the first half of it was implanted, rather than being developed by you. I cannot see how to dispute this possibility. |