9 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) |
| 4988 | Folk psychology may not be reducible, but that doesn't make it false [Kirk,R on Churchland,PM] |
| Full Idea: It may well be that completed neuroscience will not include a reduction of folk psychology, but why should that be a reason to regard it as false? It would only be a reason if irreducibility entailed that they could not possibly both be true. | |
| From: comment on Paul M. Churchland (Eliminative Materialism and Prop. Attitudes [1981]) by Robert Kirk - Mind and Body §3.9 | |
| A reaction: If all our behaviour had been explained by a future neuro-science, this might not falsify folk psychology, but it would totally marginalise it. It is still possible that dewdrops are placed on leaves by fairies, but this is no longer a hot theory. |
| 4987 | Eliminative materialism says folk psychology will be replaced, not reduced [Churchland,PM] |
| Full Idea: Eliminative materialism says our common-sense conception of psychological phenomena is a radically false theory, so defective that both the principles and the ontology of that theory will eventually be displaced (rather than reduced). | |
| From: Paul M. Churchland (Eliminative Materialism and Prop. Attitudes [1981], Intro) | |
| A reaction: It is hard to see what you could replace the idea of a 'belief' with in ordinary conversation. We may reduce beliefs to neuronal phenomena, but we can't drop the vocabulary of the macro-phenomena. The physics of weather doesn't eliminate 'storms'. |
| 7270 | Historical interpretation aims to recapture the author's view of the work [Croce] |
| Full Idea: Historical interpretation enables us to see a work of art as its author saw it in the moment of production. | |
| From: Benedetto Croce (Aesthetic as Science of Expression [1902], §II), quoted by W Wimsatt/W Beardsley - The Intentional Fallacy §II | |
| A reaction: Wimsatt and Beardsley quote this as the romantic antithesis of their own view, but there is a blurring between understanding a work and judging. Personally I consider intentions essential for understanding, and valuable for judgement. |