10 ideas
| 21697 | The Struthionic Fallacy is that of burying one's head in the sand [Quine] |
| Full Idea: The Struthionic Fallacy is that of burying one's head in the sand [which I name from the Greek for 'ostrich'] | |
| From: Willard Quine (Lecture on Nominalism [1946], §4) | |
| A reaction: David Armstrong said this is the the fallacy involved in a denial of universals. Quine is accusing Carnap and co. of the fallacy. |
| 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) |
| 21698 | All relations, apart from ancestrals, can be reduced to simpler logic [Quine] |
| Full Idea: Much of the theory of relations can be developed as a virtual theory, in which we seem to talk of relations, but can explain our notation in terms {finally] of just the logic of truth-functions, quantification and identity. The exception is ancestrals. | |
| From: Willard Quine (Lecture on Nominalism [1946], §8) | |
| A reaction: The irreducibility of ancestrals is offered as a reason for treating sets as universals. |
| 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) |
| 21696 | Nominalism rejects both attributes and classes (where extensionalism accepts the classes) [Quine] |
| Full Idea: 'Nominalism' is distinct from 'extensionalism'. The main point of the latter doctrine is rejection of properties or attributes in favour of classes. But class are universals equally with attributes, and nominalism in the defined sense rejects both. | |
| From: Willard Quine (Lecture on Nominalism [1946], §3) | |
| A reaction: Hence Quine soon settled on labelling himself as an 'extensionalist', leaving proper nominalism to Nelson Goodman. It is commonly observed that science massively refers to attributes, so they can't just be eliminated. |
| 19423 | By an 'idea' I mean not an actual thought, but the resources we can draw on to think [Leibniz] |
| Full Idea: What I mean by an idea is not a certain act of thinking, but a power or faculty such that we have an idea of a thing even if we are not thinking about it but know that we can think it when the occasion arises. | |
| From: Gottfried Leibniz (What is an Idea? [1676], p.281) | |
| A reaction: 'Idea' tends to be used in the seventeenth century to mean an actual mental event. It is because Leibniz believes in the unconscious mind that he can offer this rather different, and probably superior, notion of an 'idea'. |